Mathematics Made Easy

can you solve them???

1997 NATIONAL CONTESTS
1.1 AUSTRIA OLYMPIAD MATHEMATICS CONTESTS

Legacy

edit

Main article: List of topics named after Leonhard Euler

Recognition

edit

Euler is widely recognized as one of the greatest mathematicians of all time, and more likely than not the most prolific contributor to mathematics and science.[8] Mathematician and physicist John von Neumann called Euler "the greatest virtuoso of the period".[132] Mathematician François Arago said, "Euler calculated without any apparent effort, just as men breathe and as eagles sustain themselves in air".[133] He is generally ranked right below Carl Friedrich Gauss, Isaac Newton, and Archimedes among the greatest mathematicians of all time,[133] while some rank him as equal with them.[134] Physicist and mathematician Henri Poincaré called Euler the "god of mathematics".[135]

French mathematician André Weil noted that Euler stood above his contemporaries and more than anyone else was able to cement himself as the leading force of his era's mathematics:[132]

No mathematician ever attained such a position of undisputed leadership in all branches of mathematics, pure and applied, as Euler did for the best part of the eighteenth century.

Swiss mathematician Nicolas Fuss noted Euler's extraordinary memory and breadth of knowledge, saying:[5]

Knowledge that we call erudition was not inimical to him. He had read all the best Roman writers, knew perfectly the ancient history of mathematics, held in his memory the historical events of all times and peoples, and could without hesitation adduce by way of examples the most trifling of historical events. He knew more about medicine, botany, and chemistry than might be expected of someone who had not worked especially in those sciences.

Commemorations

edit

Euler portrait on the sixth series of the 10 Franc banknoteEuler portrait on the seventh series of the 10 Franc banknote

Euler was featured on both the sixth[136] and seventh[137] series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences.[138] The asteroid 2002 Euler was named in his honour.[139]

Integration - 💡 HINT:
Complete the square first!
x²+6x+10 = (x+3)²+1

In fluid dynamics, Euler was the first to predict the phenomenon of cavitation, in 1754, long before its first observation in the late 19th century, and the Euler number used in fluid flow calculations comes from his related work on the efficiency of turbines.[102] In 1757 he published an important set of equations for inviscid flow in fluid dynamics, that are now known as the Euler equations.[103]

Euler is well known in structural engineering for his formula giving Euler's critical load, the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness.[104]

Logic

edit

Euler is credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.[105]

An Euler diagram

An Euler diagram is a diagrammatic means of representing sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset, and disjointness). Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it.

Euler diagrams (and their refinement to Venn diagrams) were incorporated as part of instruction in set theory as part of the new math movement in the 1960s.[106] Since then, they have come into wide use as a way of visualizing combinations of characteristics.[107]

Demography

edit

In his 1760 paper A General Investigation into the Mortality and Multiplication of the Human Species Euler produced a model which showed how a population with constant fertility and mortality might grow geometrically using a difference equation. Under this geometric growth Euler also examined relationships among various demographic indices showing how they might be used to produce estimates when observations were missing. Three papers published around 150 years later by Alfred J. Lotka (1907, 1911 (with F.R. Sharpe) and 1922) adopted a similar approach to Euler's and produced their Stable Population Model. These marked the start of 20th century formal demographic modelling.[108][109][110][111][112][113]

Music

edit

One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae (Attempt at a New Theory of Music), hoping to eventually incorporate music theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.[114] Even when dealing with music, Euler's approach is mainly mathematical,[115] for instance, his introduction of binary logarithms as a way of numerically describing the subdivision of octaves into fractional parts.[116] His writings on music are not particularly numerous (a few hundred pages, in his total production of about thirty thousand pages), but they reflect an early preoccupation and one that remained with him throughout his life.[115]

A first point of Euler's musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2mA, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2m (where "m is an indefinite number, small or large, so long as the sounds are perceptible"[117]), expresses that the relation holds independently of the number of octaves concerned. The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2m.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2m.5, major third + minor sixth (C–E–C); the fourth is 2m.32, two-fourths and a tone (C–F–B♭–C); the fifth is 2m.3.5 (C–E–G–B–C); etc. Genres 12 (2m.33.5), 13 (2m.32.52) and 14 (2m.3.53) are corrected versions of the diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2m.33.52) is the "diatonico-chromatic", "used generally in all compositions",[118] and which turns out to be identical with the system described by Johann Mattheson.[119] Euler later envisaged the possibility of describing genres including the prime number 7.[120]

Euler devised a specific graph, the Speculum musicum,[121][122] to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (see above). The device drew renewed interest as the Tonnetz in Neo-Riemannian theory (see also Lattice (music)).[123]

Euler further used the principle of the "exponent" to propose a derivation of the gradus suavitatis (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he considered just intonation, i.e. 1 and only the prime numbers 3 and 5.[124] Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form ds=∑i(ki⋅pi−ki)+1 ,where pi are prime numbers and ki their exponents.[125]

Personal life

edit

On 7 January 1734, Euler married Katharina Gsell, daughter of Georg Gsell, a painter at the Academy Gymnasium in Saint Petersburg.[33] The couple bought a house by the Neva River. Three years after his wife's death in 1773,[54] Euler married her half-sister, Salome Abigail Gsell.[55] This marriage lasted until his death in 1783. Of their 13 children, five survived childhood,[56] three sons and two daughters.[57] Their first son was Johann Albrecht Euler, whose godfather was Christian Goldbach.[57] His brother Johann Heinrich settled in St. Petersburg in 1735 and was employed as a painter at the academy.[34]

Early in his life, Euler memorized Virgil's Aeneid, and by old age, he could recite the poem and give the first and last sentence on each page of the edition from which he had learnt it.[58][59] Euler knew the first hundred prime numbers and could give each of their powers up to the sixth degree.[60] Euler was known as a generous and kind person, not neurotic as seen in some geniuses, keeping his good-natured disposition even after becoming entirely blind.[60]

Eyesight deterioration

edit

Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly dying of fever,[61] he became almost blind in his right eye. Euler blamed the cartography he performed for the St. Petersburg Academy for his condition,[62] but the cause of his blindness remains the subject of speculation.[63][64] Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick called him "Cyclops". Euler said of his loss of vision, "Now I will have fewer distractions."[62] In 1766 a cataract in his left eye was discovered. Though couching of the cataract temporarily improved his vision, complications rendered him almost totally blind in the left eye as well.[39] His condition appeared to have little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased;[65] in 1775, he produced, on average, one mathematical paper per week.[39]

Death

edit

Euler's grave at the Alexander Nevsky Monastery

In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with Anders Johan Lexell when he collapsed and died of a brain hemorrhage.[63] Jacob von Staehlin [de] wrote a short obituary for the Russian Academy of Sciences and Russian mathematician Nicolas Fuss, one of Euler's disciples, wrote a more detailed eulogy,[56] which he delivered at a memorial meeting. In his eulogy for the French Academy, French mathematician and philosopher Marquis de Condorcet wrote:

.. il cessa de calculer et de vivre.

.. he ceased to calculate and to live.[66]

Euler was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island. In 1837, the Russian Academy of Sciences installed a new monument, replacing his overgrown grave plaque. In 1957, to commemorate the 250th anniversary of his birth, his tomb was moved to the Lazarevskoe Cemetery at the Alexander Nevsky Monastery.[67]

Euler's practical engineering abilities, stating:

I wanted to have a water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry![45]

However, the disappointment was almost surely unwarranted from a technical perspective. Euler's calculations look likely to be correct, even if Euler's interactions with Frederick and those constructing his fountain may have been dysfunctional.[46]

Throughout his stay in Berlin, Euler maintained a strong connection to the academy in St. Petersburg and also published 109 papers in Russia.[47] He also assisted students from the St. Petersburg academy and at times accommodated Russian students in his house in Berlin.[47] In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops.[42] Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler's estate, with Empress Elizabeth of Russia later adding a further payment of 4000 rubles—an exorbitant amount at the time.[48] Euler decided to leave Berlin in 1766 and return to Russia.[49]

During his Berlin years (1741–1766), Euler was at the peak of his productivity. He wrote 380 works, 275 of which were published.[50] This included 125 memoirs in the Berlin Academy and over 100 memoirs sent to the St. Petersburg Academy, which had retained him as a member and paid him an annual stipend. Euler's Introductio in Analysin Infinitorum was published in two parts in 1748. In addition to his own research, Euler supervised the library, the observatory, the botanical garden, and the publication of calendars and maps from which the academy derived income.[51] He was even involved in the design of the water fountains at Sanssouci, the King's summer palace.[52]

Second Saint Petersburg period (1766–1783)

edit

The political situation in Russia stabilized after Catherine the Great's accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he was assisted by his student Anders Johan Lexell.[53] While living in St. Petersburg, a fire in 1771 destroyed his home.[54]

Early life

edit

Leonhard Euler was born in Basel on 15 April 1707 to Paul III Euler, a pastor of the Reformed Church, and Marguerite (née Brucker), whose ancestors include a number of well-known scholars in the classics.[16] He was the oldest of four children, with two younger sisters, Anna Maria, and Maria Magdalena, and a younger brother, Johann Heinrich.[17][16] Soon after Leonhard's birth, the Eulers moved from Basel to Riehen, Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood.[16]

From a young age, Euler received schooling in mathematics from his father, who had taken courses from Jacob Bernoulli some years earlier at the University of Basel. Around the age of eight, Euler was sent to live at his maternal grandmother's house and enrolled in the Latin school in Basel. In addition, he received private tutoring from Johannes Burckhardt, a young theologian with a keen interest in mathematics.[16]

In 1720, at age 13, Euler enrolled at the University of Basel.[4] Attending university at such a young age was not unusual at the time.[16] The course on elementary mathematics was given by Johann Bernoulli, the younger brother of the deceased Jacob Bernoulli, who had taught Euler's father. Johann Bernoulli and Euler soon got to know each other better. Euler described Bernoulli in his autobiography:[18]

the famous professor Johann Bernoulli [...] made it a special pleasure for himself to help me along in the mathematical sciences. Private lessons, however, he refused because of his busy schedule. However, he gave me a far more salutary advice, which consisted in myself getting a hold of some of the more difficult mathematical books and working through them with great diligence, and should I encounter some objections or difficulties, he offered me free access to him every Saturday afternoon, and he was gracious enough to comment on the collected difficulties, which was done with such a desired advantage that, when he resolved one of my objections, ten others at once disappeared, which certainly is the best method of making happy progress in the mathematical sciences.

During this time, Euler, backed by Bernoulli, obtained his father's consent to become a mathematician instead of a pastor.[19][20]

In 1723, Euler received a Master of Philosophy with a dissertation that compared the philosophies of René Descartes and Isaac Newton.[16] Afterwards, he enrolled in the theological faculty of the University of Basel.[20]

In 1726, Euler completed a dissertation on the propagation of sound titled De Sono,[21][22] with which he unsuccessfully attempted to obtain a position at the University of Basel.[23] In 1727, he entered the Paris Academy prize competition (offered annually and later biennially by the academy beginning in 1720)[24] for the first time. The problem posed that year was to find the best way to place the masts on a ship. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place.[25] Over the years, Euler entered this competition 15 times,[24] winning 12 of them.[25]

Leonhard Euler (/ˈɔɪlər/ OY-lər;[b] 15 April 1707 – 18 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, music theorist and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function.[3] He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.[4] Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory".[5] He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.
Euler is credited for popularizing the Greek letter
π
{\displaystyle \pi } (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation
f
(
x
)
{\displaystyle f(x)} for the value of a function, the letter
i
{\displaystyle i} to express the imaginary unit
−
1
{\displaystyle {\sqrt {-1}}}, the Greek letter
Σ
{\displaystyle \Sigma } (capital sigma) to express summations, the Greek letter
Δ
{\displaystyle \Delta } (capital delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters.[6] He gave the current definition of the constant
e
{\displaystyle e}, the base of the natural logarithm, now known as Euler's number.[7] Euler made contributions to applied mathematics and engineering, such as his study of ships, which helped navigation; his three volumes on optics, which contributed to the design of microscopes and telescopes; and his studies of beam bending and column critical loads.[8]

Euler is credited with being the first to develop graph theory (partly as a solution for the problem of the Seven Bridges of Königsberg, which is also considered the first practical application of topology). He also became famous for, among many other accomplishments, solving several unsolved problems in number theory and analysis, including the famous Basel problem. Euler has also been credited for discovering that the sum of the numbers of vertices and faces minus the number of edges of a polyhedron that has no holes equals 2, a number now commonly known as the Euler characteristic. In physics, Euler reformulated Isaac Newton's laws of motion into new laws in his two-volume work Mechanica to better explain the motion of rigid bodies. He contributed to the study of elastic deformations of solid objects. Euler formulated the partial differential equations for the motion of inviscid fluid,[8] and laid the mathematical foundations of potential theory.[5]

Euler is regarded as arguably the most prolific contributor in the history of mathematics and science, and the greatest mathematician of the 18th century.[9][8] His 866 publications and his correspondence were collected in the Opera Omnia Leonhard Euler.[10][11][12] Several great mathematicians who worked after Euler's death have recognised his importance in the field: Pierre-Simon Laplace said, "Read Euler, read Euler, he is the master of us all";[13][c] Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it."[14][d]

Happy birthday Leonhard Euler